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I don't know of any that are. I think most Linux distros use GCC, supporting software with common licenses and one built specifically to handle GNU code. Clang is currently being embraced by the BSD community for similar reasons. (The proceeding is a huge simplification of political and technical problems)
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Chris SAug 3 '11 at 12:53

@James Please don't edit the answer into the question; you can post an actual answer below and mark it accepted. Also, if you register on SF you should automatically take ownership of the question here
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Michael Mrozek♦Aug 3 '11 at 16:09

As a late response to this question, what @jmtd said is exactly what happens periodically in the Debian archive.

Packages are recompiled with newer toolchains (compilers, linkers, etc.) and, when recompilation fails, bugs against the packages that failed to be compiled are filed with the indication of FTBFS ("failed to build from source").

Such bugs are generally an indication of something bad happening and are usually ranked with high severity and given the status of "release critical" bugs, meaning that a new release can't be done with those bugs unsolved.

In particular, Lucas Nussbaum has been recompiling the whole archive of the Debian Project in a grid as a means of some Quality Assessment.

Recently, though, Sylvestre Ledru and some other Debian Developers have mentoredstudents in Google's Summer of Code of 2012 to allow substitution of both GCC by Clang and libstdc++ (GCC's support library for C++) by libc++ (Clang/LLVM's counterpart).

There is a site where the results of compilation of the whole collection of Debian packages were published and further reports were documented by LWN and the program for decoupling the Debian build process from GCC was successfully concluded.

So we may soon see a flavour of Debian compiled with clang/llvm, depending on how much support these successful results can gain.

Whilst the end-user binaries are not built with clang/llvm, most of the Debian archive has been built (and rebuilt) with clang/llvm (and binutils-gold and other bits and pieces) by people who want to expose portability bugs in the software suite (and the compilers themselves).

Yes, and those are built with gcc. Separately from the end-user binaries, the source packages are rebuilt as part of multiple continuous-integration processes, some of which use alternative compiler/linker/libc toolchains.
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jmtdAug 30 '11 at 15:23